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Transactions of the American Mathematical Society

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Geodesic-Einstein metrics and nonlinear stabilities


Authors: Huitao Feng, Kefeng Liu and Xueyuan Wan
Journal: Trans. Amer. Math. Soc. 371 (2019), 8029-8049
MSC (2010): Primary 53C55, 53C60
DOI: https://doi.org/10.1090/tran/7658
Published electronically: December 21, 2018
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Abstract: In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.


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Additional Information

Huitao Feng
Affiliation: Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, People’s Republic of China
Email: fht@nankai.edu.cn

Kefeng Liu
Affiliation: Department of Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China; and Department of Mathematics, University of California, Los Angeles, California 90095
Email: liu@math.ucla.edu

Xueyuan Wan
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology,University of Gothenberg, 41296 Gothenburg, Sweden
Email: xwan@chalmers.se

DOI: https://doi.org/10.1090/tran/7658
Received by editor(s): April 18, 2017
Received by editor(s) in revised form: June 26, 2018
Published electronically: December 21, 2018
Additional Notes: The first author was partially supported by NSFC (Grants No. 11221091, 11271062, 11571184) and the Fundamental Research Funds for the Central Universities
The second author was partially supported by NSF (Grant No. 1510216).
Article copyright: © Copyright 2018 American Mathematical Society